We present alternative expressions for modularity in graphs. Modularity is used as a measure to characterize the community of networks, which is one of the most important features in real-world networks, especially social networks.

In this paper, a class of infinite sequences with positive integer terms is considered. If the Twin Prime Conjecture, stipulating that there are infinitely many twin prime numbers, is true, then the sequence of all twin prime numbers belongs to the same class. In the present investigation, it is proved that the mentioned class is non-empty and moreover there exists at least one element of this class containing all twin primes.

Integer structure analysis illustrates the critical structural factors which underpin the failure of (N4m + M4m) ever to equal an equivalent power. The number 3 plays a vital role as integers divisible by 3, when raised to an even power of the form 4m, have rows in a table of modular rings which are triangular numbers, whereas other integers raised to the same power have rows which are pentagonal numbers. The substructure within these sequences of pentagonal numbers is order within order, analogous to structure in chaos theory.

An integer structure analysis (ISA) of the triangular, tetrahedral, pentagonal and pyramidal numbers is developed. The relationships among the elements and the powers of the elements of these sequences are discussed. In particular, the triangular and pentagonal numbers are directly linked and are structurally important for the formation of triples. The class structure in the modular rings Z3 and Z4 of some elements of these sequences reinforce previous studies of their properties.

We introduce an arithmetic function and study some of its properties analogous to Möbius function in analytic number theory. In addition, we derive some simple expressions to connect infinite series and infinite products through this new arithmetic function. Moreover, we study applications of this new function to derive simple identities on partition of an integer and some special infinite products in terms of multiplicative functions.

Let Γ be an abelian group and B be a subset of Γ. The addition Cayley graph G′ = Cay+(Γ, B) is the graph having the vertex set V (G′) = Γ and the edge set E(G′) = {ab : a + b ∈ B}, where a, b ∈ Γ. For a positive integer n > 1, the unitary addition Cayley graph Gn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a + b ∈ Un. The unitary addition Cayley graph Gn is also defined as, Gn = Cay+(Zn, Un). In this paper, we discuss the several properties of unitary addition Cayley graphs and also obtain the characterization of planarity and outerplanarity of unitary addition Cayley graphs.